CLEVELAND — Like pushing on an open door, the Giants handled the winless Cleveland Browns on Sunday afternoon, with wide receiver Odell Beckham Jr. overcoming a thumb injury early on to catch six passes for 96 yards and two touchdowns in a 27-13 victory.

It was the sixth straight win for the Giants (8-3). If the season had ended Sunday afternoon, they would have earned the first wild-card spot in the N.F.C. They travel to face the Pittsburgh Steelers (6-5) next Sunday.

The Browns continued their run into the wrong history books, dropping to 0-12, with a league-worst point differential of minus-155. If they lose their final four games, they will become the second 0-16 team in N.F.L. history, joining the 2008 Detroit Lions. They have a bye next weekend.

“We’ve got to find a way to put together a win,” said Hue Jackson, the Browns’ first-year head coach. “And who knows? There’s four games that we’ve got an opportunity to go out and play, so there’s four opportunities to win, and there’s another four opportunities to lose.”

The game was closer than the final score might make it seem. It was 0-0 until midway through the second quarter, and the Giants’ lead was cut to 7 points in the fourth. A crucial Giants score came on a turnover, with defensive lineman Jason Pierre-Paul recovering a fumble by Browns quarterback Josh McCown and returning it 43 yards.

Giants Coach Ben McAdoo bemoaned errors, including nine penalties, which cumulatively cost a football field — 100 yards. A holding penalty on a punt negated what would have been a long touchdown return by Beckham.

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Giants wide receiver Dwayne Harris scoring in front of Browns cornerback Briean Boddy-Calhoun during the second quarter on a 13-yard pass from Eli Manning.CreditGregory Shamus/Getty Images

“We had some opportunities to make some big plays and we didn’t make them,” McAdoo said.

Giants quarterback Eli Manning pointed to “too many three and outs” and a need to “be better on third down” — the Giants converted 4 of 13.

“We’ve just got to find ways to not let the defense back on the field, close these games out ourselves, and take steps as an offense,” Beckham said. “I don’t think we took a step back today, but I do think we could’ve taken a bigger step forward.”

Cleveland’s offense, with its rotating cast of quarterbacks, has been an ostentatious liability this season, but its defense has more shameful statistics. It entered the game ranked second-worst in the N.F.L. in both yards and scoring per game.

But with two Giants touchdowns coming either directly or indirectly off turnovers Sunday, there was a prevailing sense that Cleveland’s offense had let its defense down.

“The way our defense played today, you can win a game that way,” said Browns wide receiver Terrelle Pryor Sr., who had six catches for 131 yards. “Our defense,” he added, “has played really well the past three games. Our offense put them in bad situations. We need to start scoring points. It’s as simple as that.”

The Giants’ first score came after Browns running back Isaiah Crowell fumbled on Cleveland’s 25-yard line midway through the second quarter. Manning found wide receiver Dwayne Harris in the end zone on a 13-yard fade — Harris’s first catch of the season.

The Giants’ Bobby Rainey, handling return duties ordinarily reserved for Harris, later muffed a punt, turning the ball over on the Giants’ 30. Cleveland ended up kicking a field goal at the 2-yard line, one of two from inside the 10-yard line.

With 1 minute 38 seconds left in the first half, the Giants finally found an offensive rhythm, with a four-play, 69-yard drive that culminated in Manning finding Beckham on a short crossing route. Beckham outran the Browns’ top cornerback, Joe Haden, then jetted upfield to complete a 32-yard touchdown.

The Giants’ offense stalled again in the second half, but the Browns bailed them out. Pinned at the Cleveland 5-yard line after another Giants punt, McCown hit Pryor for a 54-yard catch on a play-action pass. Three snaps later, McCown dropped back to pass and was hit as he threw. The ball popped into the air and fell right into the arms of Pierre-Paul, who scored. A missed extra-point attempt made it 20-6.

It was Pierre-Paul’s third career touchdown, and the first since a Fourth of July fireworks accident in 2015 severely injured his right hand, leading to the amputation of his right index finger. The turnover was originally ruled an interception but later defined as a fumble. “The ball never touched the ground, so we’ll call it an interception,” Pierre-Paul said after the game with a smile.

The Browns made things interesting on the drive that followed, again marching up the field on the strength of McCown’s arm. He found the rookie wide receiver Corey Coleman wide open in the end zone to make it 20-13.

But the Giants responded with their own big drive. Manning found Beckham in the middle of the field for a 41-yard catch-and-run. Then Manning flipped the ball to Sterling Shepard for 22 yards to get the Giants to the Browns’ 6-yard line. There, on third down, he found Beckham in the end zone on a short cross.

Manning finished 15 of 27 for 194 yards, three touchdowns and no interceptions.

The Giants, who entered the game with the second-fewest yards rushing per game in the N.F.L., again struggled on the ground. Running backs Rashad Jennings and Paul Perkins combined for 84 yards on 24 carries.

Throughout the game, the Giants’ defensive line attacked the Browns’ blockers, collapsing what might have existed of McCown’s pocket. The Giants finished with seven sacks and eight tackles for a loss. At times, the Browns played competently enough to belie their record. Aside from the fumble, Crowell had 22 touches, between rushes and receptions, for 91 yards. McCown finished 25 of 43 for 322 yards and a touchdown.

But at other moments, the Browns were a study in haplessness. Midway through the third quarter, Duke Johnson, with the sun behind him, allowed a punt to bounce in front of him and then past him, totaling 58 yards before he picked it up and was summarily gang-tackled for an additional loss of 3 yards. It was the kind of play one rarely sees in the N.F.L.

In the upper reaches of FirstEnergy Stadium, the fans — whose announced number was 60,034 — were outnumbered by empty burnt orange seats. Boos accompanied some of the Browns’ more dispiriting plays, including their final offensive snap, when Pierre-Paul sacked McCown, forcing a fumble the Giants recovered.

A version of this article appears in print on , on Page D1 of the New York edition with the headline: Giants, After a Slow Start, Add to the Browns’ Misery. Order Reprints | Today’s Paper | Subscribe